In this paper, we consider the generalized Camassa–Holm equation u_t + 2kux - u_xxt + au²u_x=2u_xu_xx + uu_xxx. Under substitution ξ x = ct, some new explicit periodic wave solutions and their limit forms are presented through some special phase orbits. These periodic wave solutions tend to in?nity on ξ- u plane periodically. Thus we call them periodic blow-up solutions. To our knowledge, such periodic blow-up solutions have not been found in any other equations.